Fossil fuels are only 'cheaper' because governments have become irredeemably corrupted by fossil fuel interests, so none have to face public liabilities for the damage caused by their products. The apparent 'cheapness' is a consequence only of fraudulent accounting. This is a specific case of the more general 'Pollution Paradox', viz. "The more polluting a company is, the more money it must spend on politics to ensure it is not regulated out of existence" (George Monbiot).
This makes 2 assumptions which I find to be false.
1) Any drop in the standard of living is bad. e.g. I don't need to be freezing in a store in the middle of summer.
2) Less energy use necessitates a drop in the standard of living. e.g. using energy more efficiently via lights and insulation.
2) Efficiency gains can help, but shaving off some excess here and there will not prevent a climate catastrophe. It will take drastically less fossil fuel use. I don't see that happening unless technology makes renewables cheaper than fossils. It's hard, but it's possible.
So why aren't we using nuclear?
We could conceivably swap out carbon for nuclear though. Just not carbon for [more expensive energy source]. People get antsy even when economic growth slows. A drop in the standard of living is political suicide.
I don't think we're very well psychologically equipped to deal with this.
Nuclear power is literally orders of magnitude cleaner/safer than anything else, and yet the fear mongering continues and the idealistic environmentalists are pumping more CO2 into the atmosphere with every NG turbine installed to backup the wind/solar plants. Just the other day I heard a talking head repeatedly referring to "the old dirty coal and nuke plants".
This is what I was alluding to about technology. If we can make clean energy significantly cheaper to use at scale than fossil fuels, then the world will use it.
More realistically, we need to make it cheaper to build new renewables than to keep using existing fossil fuel sources.